I have called my paper “The Logic of Inductive Inference.” It might just as well have been called “On making sense of figures.” Read by PROFESSOR PAUL CONSTANTINE and NARAKEET before the Research Group of PROFESSOR ART OWEN on Friday, January 27th, 2023 

MY CONTEXT [The rise of “Scientific Machine Learning”] [In The Logic of Scientific Discovery, Popper says induction isn’t logical, and I think I agree.] 

FISHER’S CONTEXT • Read to the Royal Statistical Society, 1934 • Summarizes 15 years of Fisher’s work - On the Mathematical Foundations of Theoretical Statistics (Phil. Trans. 1921) - Statistical Methods for Research Workers (1925) • Fisher’s complicated recognition 

Yates, The Influence of ‘Statistical Methods for Research Workers’ on the Development of the Science of Statistics (JASA 1951): It is now twenty-five years since R. A. Fisher's Statistical Methods for Research Workers was first published. These twenty-five years have seen a complete revolution in the statistical methods employed in scientific research, a revolution which can be directly attributed to the ideas contained in this book, and which has spread in ever- widening circles until there is no field of statistics in which the influence of Fisherian ideas is not profoundly felt. FISHER’S CONTEXT • Read to the Royal Statistical Society, 1934 • Summarizes 15 years of Fisher’s work - On the Mathematical Foundations of Theoretical Statistics (Phil. Trans. 1921) - Statistical Methods for Research Workers (1925) • Fisher’s complicated recognition 

A ROYAL STATISTICAL SNIPE-FEST 

Professor A. L. Bowley (the harsh critic) It is not the custom, when the Council invites a member to propose a vote of thanks on a paper, to instruct him to bless it. If to some extent I play the inverse rôle of Balaam, it is not without precedent; speakers after me can take the parts of the ass that reproved the prophet, the angel that instructed him, and the king who offered him rewards; and on that understanding I will proceed to deal with some parts of the paper. 

Professor A. L. Bowley (the harsh critic) It is not the custom, when the Council invites a member to propose a vote of thanks on a paper, to instruct him to bless it. If to some extent I play the inverse rôle of Balaam, it is not without precedent; speakers after me can take the parts of the ass that reproved the prophet, the angel that instructed him, and the king who offered him rewards; and on that understanding I will proceed to deal with some parts of the paper. 

Reply from Fisher The acerbity, to use no stronger term, with which the customary vote of thanks has been moved and seconded, strange as it must seem to visitors not familiar with our Society, does not, I confess, surprise me. From the fact that thirteen years have elapsed between the publication, by the Royal Society, of my first rough outline of the developments, which are the subjects of today's discussion, and the occurrence of that discussion itself, it is a fair inference that some at least of the Society's authorities on matters theoretical viewed these developments with disfavour, and admitted them with reluctance. The choice of order in speaking, which puzzles Professor Bowley, seems to me admirably suited to give a cumulative impression of diminishing animosity, an impression which I should be glad to see extrapolated. Professor A. L. Bowley (the harsh critic) It is not the custom, when the Council invites a member to propose a vote of thanks on a paper, to instruct him to bless it. If to some extent I play the inverse rôle of Balaam, it is not without precedent; speakers after me can take the parts of the ass that reproved the prophet, the angel that instructed him, and the king who offered him rewards; and on that understanding I will proceed to deal with some parts of the paper. 

Reply from Fisher … I find that Professor Bowley is offended with me for “introducing misleading ideas.” He does not, however, find it necessary to demonstrate that any such idea is, in fact, misleading. It must be inferred that my real crime, in the eyes of his academic eminence, must be that of “introducing ideas.” Professor A. L. Bowley (the harsh critic) The chief problem of the earlier part of the paper … lies in pp. 42 (foot) to 46. I found the treatment to be very obscure. I took it as a week-end problem, and first tried it as an acrostic, but I found that I could not satisfy all the “lights.” I tried it then as a cross-word puzzle, but I have not the facility of Sir Josiah Stamp for solving such conundrums. Next I took it as an anagram, remembering that Hooke stated his law of elasticity in that form, but when I found that there were only two vowels to eleven consonants, some of which were Greek capitals, I came to the conclusion that it might be Polish or Russian, and therefore best left to Dr. Neyman or Dr. Isserlis. Finally, I thought it must be a cypher, and after a great deal of investigation, decided that Professor Fisher had hidden the key in former papers, as is his custom, and I gave it up. But in so doing I remembered that Professor Edgeworth had written a good deal on a kindred subject, and I turned to his studies. 

Reply from Fisher … I find Dr. Isserlis using phrases from my writings as though he were expostulating with me. … I shall await with interest the results of a search, if he is willing to make one, for a prior use of this method. Dr. Isserlis There is no doubt in my mind at all about that, but Professor Fisher, like other fond parents, may perhaps see in his offspring qualities which to his mind no other children possess; others, however, may consider that the offspring are not unique. 

Reply from Fisher In reply to Dr. Irwin I should like to say that when I read the valuable summaries of recent work on Mathematical Statistics which he compiles for the Society from year to year, I am quite sure that nothing in my paper would have offered any difficulty to him, even if he had not been one of those who for years had been familiar with the fundamental processes and ideas discussed. Dr. Irwin [Dr. Irwin] happened recently to be reading that classical old book, Todhunter's History of the Theory of Probability, in which he came across the following passage. “Dr. Bowditch himself was accustomed to remark, ‘Whenever I meet in Laplace with the words, “Thus it plainly appears,” I am sure that hours, and perhaps days of hard study will alone enable me to discover how it plainly appears.’” 

Professor Greenwood (Society President) In the first place, he suspected that Professor Fisher's nomenclature had not been very helpful to the layman. He imagined that Professor Fisher recoiled from the Victorian practice of coining Greek vocables---a practice which gave occasion for a cruel practical jest in a sister learned society. But perhaps the introduction of what rude people called “gibberish” was less confusing than attaching particular meanings to words well established in the current speech of educated people. It did not, perhaps, give people much difficulty to distinguish between variance in the sense of the second moment coefficient and in the more usual sense of the attitude of any one mathematical statistician to any other mathematical statistician. But a confusion between statistics as the object of their pious founders and as a Fisherian plural was more troublesome. This, however, was only a trifle. The Galton Professor might surely claim the right exercised by Humpty Dumpty. 

Professor Greenwood (Society President) More serious was Professor Fisher’s extreme reluctance to bore his readers- surely a defect rare in statisticians. He seemed to be a little over-anxious not to incur the sneer of -- whom? -- perhaps of some of the speakers that evening---that something he had said was “obvious” or “self-evident.” He was in a little too much danger of dichotomizing his public into a tiny class of persons who were his intellectual peers, and a much larger class of persons who were to behave like the gallant six hundred. Reply from Fisher To state these objections is, of course, different from detecting the logical error in the argument on which the method is supposed to be justified; but to do this it would be necessary for that argument to be set out explicitly. 

FUNDAMENTAL CONCEPTS 

… a mathematical quantity of a different kind, which I have termed mathematical likelihood, appears to take [the place of probability] as a measure of rational belief when we are reasoning from the sample to the population. Mathematical likelihood makes its appearance in the particular kind of logical situation which I have termed a problem of estimation. … In a problem of estimation we start with a knowledge of the mathematical form of the population sampled, but without knowledge of the values of one or more parameters which enter into this form, which values would be required for the complete specification of the population; … likelihood is defined merely as a function of these parameters proportional to [the probability of the observations]. CONCEPTS • likelihood • parameter estimation 

… we are concerned with the theory of large samples, using this term, as is usual, to mean that nothing that we say shall be true, except in the limit when the size of the sample is indefinitely increased; a limit, obviously, never attained in practice. This part of the theory, to set off against the complete unreality of its subject-matter, exploits the advantage that in this unreal world all the possible merits of an estimate may be judged exclusively from its variability, or sampling variance. CONCEPTS • likelihood • parameter estimation • asymptotics • sample variance 

… we may distinguish consistent from inconsistent estimates. An inconsistent estimate is an estimate of something other than that which we want an estimate of. CONCEPTS • likelihood • parameter estimation • asymptotics • sample variance • consistency 

… we may now confine our attention to the class of estimates which, as the sample is increased without limit, tend to be distributed about their limiting value in the normal distribution. … The mean determines the bias of our estimate, and the variance determines its precision. CONCEPTS • likelihood • parameter estimation • asymptotics • sample variance • consistency • asymptotic normality 

In the cases which we are considering the variance falls off with increasing size of sample always ultimately in inverse proportion to n. The criterion of efficiency is that the limiting value of nV, where V stands for the variance of our estimate, shall be as small as possible. CONCEPTS • likelihood • parameter estimation • asymptotics • sample variance • consistency • asymptotic normality • efficiency 

We shall come later to regard i as the amount of information supplied by each of our observations, and the inequality as a statement that the reciprocal of the variance, or the invariance of the estimate, cannot exceed the amount of information in the sample. … there really are I and no less units of information to be extracted from the data, if we equate the information extracted to the invariance of our estimate. 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 i = X all possible obs. ( 1 f ✓ @f @✓ ◆2 ) AAACCHicbZDLSgMxFIbP1Futt6pLFwaL4ELKjHgpFKHgRncV7AU6pWTSTBuayYxJRijDLN34Km5cKOLWR3Dn25heFtr6Q+DjP+dwcn4v4kxp2/62MguLS8sr2dXc2vrG5lZ+e6euwlgSWiMhD2XTw4pyJmhNM81pM5IUBx6nDW9wNao3HqhULBR3ehjRdoB7gvmMYG2sTn7f9SUmiZMm9dQtu5zeu2UkGHLLlwZujjv5gl20x0Lz4EyhAFNVO/kvtxuSOKBCE46Vajl2pNsJlpoRTtOcGysaYTLAPdoyKHBAVTsZH5KiQ+N0kR9K84RGY/f3RIIDpYaBZzoDrPtqtjYy/6u1Yu2X2gkTUaypIJNFfsyRDtEoFdRlkhLNhwYwkcz8FZE+Nslok13OhODMnjwP9ZOic148uz0tVErTOLKwBwdwBA5cQAWuoQo1IPAIz/AKb9aT9WK9Wx+T1ow1ndmFP7I+fwBRq5gz 1 V  ni = I, CONCEPTS • likelihood • parameter estimation • asymptotics • sample variance • consistency • asymptotic normality • efficiency • Fisher information • invariance (??) 

We shall come later to regard i as the amount of information supplied by each of our observations, and the inequality as a statement that the reciprocal of the variance, or the invariance of the estimate, cannot exceed the amount of information in the sample. … there really are I and no less units of information to be extracted from the data, if we equate the information extracted to the invariance of our estimate. AAACY3icbZFdaxQxFIYzo9Y6VTut3okQXIQKssyUWgtFKHjjZQW3LWzW5Uz2zG5oZjIkZ8Q1zJ/0zjtv/B9mdwfU1gOBh/d8JW+KRitHWfYjiu/cvbd1f/tBsvPw0ePddG//wpnWShxJo429KsChVjWOSJHGq8YiVIXGy+L6/Sp/+QWtU6b+RMsGJxXMa1UqCRSkafpNcXH6TpwmwrXV1AvCr+RBa94Y51SYwk3hhl2XCI0lCZ+I0oL0eefLTrzeqAe9KBqwpEDzsvvDghZIEPqtmi/o1efDnkQ3TQfZMFsHvw15DwPWx/k0/S5mRrYV1iQ1ODfOs4YmfrVIagwrWocNyGuY4zhgDRW6iV971PGXQZnx0thwauJr9e8OD5Vzy6oIlRXQwt3MrcT/5cYtlScTr+qmJazlZlHZak6GrwznM2VRkl4GAGlVuCuXCwh+UfiWJJiQ33zybbg4HObHwzcfjwZnJ70d2+wZe8EOWM7esjP2gZ2zEZPsZ7QV7UZp9Cveiffjp5vSOOp7nrB/In7+G+qyt0k= i = X all possible obs. ( 1 f ✓ @f @✓ ◆2 ) AAACCHicbZDLSgMxFIbP1Futt6pLFwaL4ELKjHgpFKHgRncV7AU6pWTSTBuayYxJRijDLN34Km5cKOLWR3Dn25heFtr6Q+DjP+dwcn4v4kxp2/62MguLS8sr2dXc2vrG5lZ+e6euwlgSWiMhD2XTw4pyJmhNM81pM5IUBx6nDW9wNao3HqhULBR3ehjRdoB7gvmMYG2sTn7f9SUmiZMm9dQtu5zeu2UkGHLLlwZujjv5gl20x0Lz4EyhAFNVO/kvtxuSOKBCE46Vajl2pNsJlpoRTtOcGysaYTLAPdoyKHBAVTsZH5KiQ+N0kR9K84RGY/f3RIIDpYaBZzoDrPtqtjYy/6u1Yu2X2gkTUaypIJNFfsyRDtEoFdRlkhLNhwYwkcz8FZE+Nslok13OhODMnjwP9ZOic148uz0tVErTOLKwBwdwBA5cQAWuoQo1IPAIz/AKb9aT9WK9Wx+T1ow1ndmFP7I+fwBRq5gz 1 V  ni = I, CONCEPTS • likelihood • parameter estimation • asymptotics • sample variance • consistency • asymptotic normality • efficiency • Fisher information • invariance (??) 

In certain cases estimates are shown to exist such that, when they are given, the distributions of all other estimates are independent of the parameter require. Such estimates, which are called sufficient, contain, even from finite samples, the whole of the information supplied by the data. CONCEPTS • likelihood • parameter estimation • asymptotics • sample variance • consistency • asymptotic normality • efficiency • Fisher information • invariance (??) • sufficiency 

FISHER’S CONCLUSION 

If we are satisfied of the logical soundness of the criteria developed, we are in a position to apply them to test the claim that mathematical likelihood supplies, in the logical situation prevailing in problems of estimation, a measure of rational belief analogous to, though mathematically different from, that supplied by mathematical probability in those problems of uncertain deductive inference for which the theory of probability was developed. This claim may be substantiated by two facts. First, that the particular method of estimation, arrived at by choosing those values of the parameters the likelihood of which is greatest, is found to elicit not less information than any other method which can be adopted. Secondly, the residual information supplied by the sample, which is not included in a mere statement of the parametric values which maximize the likelihood, can be obtained from other characteristics of the likelihood function; such as, if it is differentiable, its second and higher derivatives at the maximum. Thus, basing our theory entirely on considerations independent of the possible relevance of mathematical likelihood to inductive inferences in problems of estimation, we seem inevitably led to recognize in this quantity the medium by which all such information as we possess may be appropriately conveyed. NOTES 

If we are satisfied of the logical soundness of the criteria developed, we are in a position to apply them to test the claim that mathematical likelihood supplies, in the logical situation prevailing in problems of estimation, a measure of rational belief analogous to, though mathematically different from, that supplied by mathematical probability in those problems of uncertain deductive inference for which the theory of probability was developed. This claim may be substantiated by two facts. First, that the particular method of estimation, arrived at by choosing those values of the parameters the likelihood of which is greatest, is found to elicit not less information than any other method which can be adopted. Secondly, the residual information supplied by the sample, which is not included in a mere statement of the parametric values which maximize the likelihood, can be obtained from other characteristics of the likelihood function; such as, if it is differentiable, its second and higher derivatives at the maximum. Thus, basing our theory entirely on considerations independent of the possible relevance of mathematical likelihood to inductive inferences in problems of estimation, we seem inevitably led to recognize in this quantity the medium by which all such information as we possess may be appropriately conveyed. NOTES • Logical soundness or suitability? 

If we are satisfied of the logical soundness of the criteria developed, we are in a position to apply them to test the claim that mathematical likelihood supplies, in the logical situation prevailing in problems of estimation, a measure of rational belief analogous to, though mathematically different from, that supplied by mathematical probability in those problems of uncertain deductive inference for which the theory of probability was developed. This claim may be substantiated by two facts. First, that the particular method of estimation, arrived at by choosing those values of the parameters the likelihood of which is greatest, is found to elicit not less information than any other method which can be adopted. Secondly, the residual information supplied by the sample, which is not included in a mere statement of the parametric values which maximize the likelihood, can be obtained from other characteristics of the likelihood function; such as, if it is differentiable, its second and higher derivatives at the maximum. Thus, basing our theory entirely on considerations independent of the possible relevance of mathematical likelihood to inductive inferences in problems of estimation, we seem inevitably led to recognize in this quantity the medium by which all such information as we possess may be appropriately conveyed. NOTES • Logical soundness or suitability? • Likelihood measures rational belief just like probability does for Bayesian analysis 

If we are satisfied of the logical soundness of the criteria developed, we are in a position to apply them to test the claim that mathematical likelihood supplies, in the logical situation prevailing in problems of estimation, a measure of rational belief analogous to, though mathematically different from, that supplied by mathematical probability in those problems of uncertain deductive inference for which the theory of probability was developed. This claim may be substantiated by two facts. First, that the particular method of estimation, arrived at by choosing those values of the parameters the likelihood of which is greatest, is found to elicit not less information than any other method which can be adopted. Secondly, the residual information supplied by the sample, which is not included in a mere statement of the parametric values which maximize the likelihood, can be obtained from other characteristics of the likelihood function; such as, if it is differentiable, its second and higher derivatives at the maximum. Thus, basing our theory entirely on considerations independent of the possible relevance of mathematical likelihood to inductive inferences in problems of estimation, we seem inevitably led to recognize in this quantity the medium by which all such information as we possess may be appropriately conveyed. NOTES • Logical soundness or suitability? • Likelihood measures rational belief just like probability does for Bayesian analysis • How does likelihood measure belief? 1) max likelihood gets the most information from the sample 

If we are satisfied of the logical soundness of the criteria developed, we are in a position to apply them to test the claim that mathematical likelihood supplies, in the logical situation prevailing in problems of estimation, a measure of rational belief analogous to, though mathematically different from, that supplied by mathematical probability in those problems of uncertain deductive inference for which the theory of probability was developed. This claim may be substantiated by two facts. First, that the particular method of estimation, arrived at by choosing those values of the parameters the likelihood of which is greatest, is found to elicit not less information than any other method which can be adopted. Secondly, the residual information supplied by the sample, which is not included in a mere statement of the parametric values which maximize the likelihood, can be obtained from other characteristics of the likelihood function; such as, if it is differentiable, its second and higher derivatives at the maximum. Thus, basing our theory entirely on considerations independent of the possible relevance of mathematical likelihood to inductive inferences in problems of estimation, we seem inevitably led to recognize in this quantity the medium by which all such information as we possess may be appropriately conveyed. NOTES • Logical soundness or suitability? • Likelihood measures rational belief just like probability does for Bayesian analysis • How does likelihood measure belief? 1) max likelihood gets the most information from the sample 2) provides measures of uncertainty 

If we are satisfied of the logical soundness of the criteria developed, we are in a position to apply them to test the claim that mathematical likelihood supplies, in the logical situation prevailing in problems of estimation, a measure of rational belief analogous to, though mathematically different from, that supplied by mathematical probability in those problems of uncertain deductive inference for which the theory of probability was developed. This claim may be substantiated by two facts. First, that the particular method of estimation, arrived at by choosing those values of the parameters the likelihood of which is greatest, is found to elicit not less information than any other method which can be adopted. Secondly, the residual information supplied by the sample, which is not included in a mere statement of the parametric values which maximize the likelihood, can be obtained from other characteristics of the likelihood function; such as, if it is differentiable, its second and higher derivatives at the maximum. Thus, basing our theory entirely on considerations independent of the possible relevance of mathematical likelihood to inductive inferences in problems of estimation, we seem inevitably led to recognize in this quantity the medium by which all such information as we possess may be appropriately conveyed. NOTES • Logical soundness or suitability? • Likelihood measures rational belief just like probability does for Bayesian analysis • How does likelihood measure belief? 1) max likelihood gets the most information from the sample 2) provides measures of uncertainty • Likelihood communicates information 

If we are satisfied of the logical soundness of the criteria developed, we are in a position to apply them to test the claim that mathematical likelihood supplies, in the logical situation prevailing in problems of estimation, a measure of rational belief analogous to, though mathematically different from, that supplied by mathematical probability in those problems of uncertain deductive inference for which the theory of probability was developed. This claim may be substantiated by two facts. First, that the particular method of estimation, arrived at by choosing those values of the parameters the likelihood of which is greatest, is found to elicit not less information than any other method which can be adopted. Secondly, the residual information supplied by the sample, which is not included in a mere statement of the parametric values which maximize the likelihood, can be obtained from other characteristics of the likelihood function; such as, if it is differentiable, its second and higher derivatives at the maximum. Thus, basing our theory entirely on considerations independent of the possible relevance of mathematical likelihood to inductive inferences in problems of estimation, we seem inevitably led to recognize in this quantity the medium by which all such information as we possess may be appropriately conveyed. NOTES • Logical soundness or suitability? • Likelihood measures rational belief just like probability does for Bayesian analysis • How does likelihood measure belief? 1) max likelihood gets the most information from the sample 2) provides measures of uncertainty • Likelihood communicates information There are two reasons that likelihood is a measure of rational belief in parameter estimation problems: 1. maximum likelihood captures the most information, 2. likelihood properties let us reason about uncertainty in the parameter estimates. Therefore, likelihood is a surrogate for information---even beyond the context of inductive inference. Paraphrase: 

INDUCTION / DEDUCTION Uncertainty and Rigo(u)r 

... I welcomed also the invitation, personally, as affording an opportunity of putting forward the opinion to which I find myself more and more strongly drawn, that the essential effect of the general body of researches in mathematical statistics during the last fifteen years is fundamentally a reconstruction of logical rather than mathematical ideas, although the solution of mathematical problems has contributed essentially to this reconstruction. NOTES CONTEXT: Introduction 

... I welcomed also the invitation, personally, as affording an opportunity of putting forward the opinion to which I find myself more and more strongly drawn, that the essential effect of the general body of researches in mathematical statistics during the last fifteen years is fundamentally a reconstruction of logical rather than mathematical ideas, although the solution of mathematical problems has contributed essentially to this reconstruction. NOTES • Logic not math (??) CONTEXT: Introduction 

I have called my paper “The Logic of Inductive Inference.” It might just as well have been called “On making sense of figures.” For everyone who does habitually attempt the difficult task of making sense of figures is, in fact, essaying a logical process of the kind we call inductive, in that he is attempting to draw inferences from the particular to the general; or, as we more usually say in statistics, from the sample to the population. NOTES • Logic not math (??) CONTEXT: Introduction 

I have called my paper “The Logic of Inductive Inference.” It might just as well have been called “On making sense of figures.” For everyone who does habitually attempt the difficult task of making sense of figures is, in fact, essaying a logical process of the kind we call inductive, in that he is attempting to draw inferences from the particular to the general; or, as we more usually say in statistics, from the sample to the population. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning CONTEXT: Introduction 

I have called my paper “The Logic of Inductive Inference.” It might just as well have been called “On making sense of figures.” For everyone who does habitually attempt the difficult task of making sense of figures is, in fact, essaying a logical process of the kind we call inductive, in that he is attempting to draw inferences from the particular to the general; or, as we more usually say in statistics, from the sample to the population. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general CONTEXT: Introduction 

Such inferences we recognize to be uncertain inferences, but it does not follow from this that they are not mathematically rigorous inferences. In the theory of probability we are habituated to statements which may be entirely rigorous, involving the concept of probability, which, if translated into verifiable observations, have the character of uncertain statements. They are rigorous because they contain within themselves an adequate specification of the nature and extent of the uncertainty involved. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general CONTEXT: Introduction 

Such inferences we recognize to be uncertain inferences, but it does not follow from this that they are not mathematically rigorous inferences. In the theory of probability we are habituated to statements which may be entirely rigorous, involving the concept of probability, which, if translated into verifiable observations, have the character of uncertain statements. They are rigorous because they contain within themselves an adequate specification of the nature and extent of the uncertainty involved. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general • Inferences can be both uncertain and rigorous CONTEXT: Introduction 

Such inferences we recognize to be uncertain inferences, but it does not follow from this that they are not mathematically rigorous inferences. In the theory of probability we are habituated to statements which may be entirely rigorous, involving the concept of probability, which, if translated into verifiable observations, have the character of uncertain statements. They are rigorous because they contain within themselves an adequate specification of the nature and extent of the uncertainty involved. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general • Inferences can be both uncertain and rigorous • Rigor involves adequately specifying uncertainty (??) CONTEXT: Introduction 

This distinction between uncertainty and lack of rigour, which should be familiar to all students of the theory of probability, seems not to be widely understood by those mathematicians who have been trained, as most mathematicians are, almost exclusively in the technique of deductive reasoning; indeed, it would not be surprising or exceptional to find mathematicians of this class ready to deny at first sight that rigorous inferences from the particular to the general were even possible. That they are, in fact, possible is, I suppose, recognized by all who are familiar with the modern work. It will be sufficient here to note that the denial implies, qualitatively, that the process of learning by observation, or experiment, must always lack real cogency. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general • Inferences can be both uncertain and rigorous • Rigor involves adequately specifying uncertainty (??) CONTEXT: Introduction 

This distinction between uncertainty and lack of rigour, which should be familiar to all students of the theory of probability, seems not to be widely understood by those mathematicians who have been trained, as most mathematicians are, almost exclusively in the technique of deductive reasoning; indeed, it would not be surprising or exceptional to find mathematicians of this class ready to deny at first sight that rigorous inferences from the particular to the general were even possible. That they are, in fact, possible is, I suppose, recognized by all who are familiar with the modern work. It will be sufficient here to note that the denial implies, qualitatively, that the process of learning by observation, or experiment, must always lack real cogency. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general • Inferences can be both uncertain and rigorous • Rigor involves adequately specifying uncertainty (??) • For mathematicians trained in deduction, uncertainty arises from a lack of rigor CONTEXT: Introduction 

This distinction between uncertainty and lack of rigour, which should be familiar to all students of the theory of probability, seems not to be widely understood by those mathematicians who have been trained, as most mathematicians are, almost exclusively in the technique of deductive reasoning; indeed, it would not be surprising or exceptional to find mathematicians of this class ready to deny at first sight that rigorous inferences from the particular to the general were even possible. That they are, in fact, possible is, I suppose, recognized by all who are familiar with the modern work. It will be sufficient here to note that the denial implies, qualitatively, that the process of learning by observation, or experiment, must always lack real cogency. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general • Inferences can be both uncertain and rigorous • Rigor involves adequately specifying uncertainty (??) • For mathematicians trained in deduction, uncertainty arises from a lack of rigor • Induction cannot be rigorous. (I think this!) CONTEXT: Introduction 

This distinction between uncertainty and lack of rigour, which should be familiar to all students of the theory of probability, seems not to be widely understood by those mathematicians who have been trained, as most mathematicians are, almost exclusively in the technique of deductive reasoning; indeed, it would not be surprising or exceptional to find mathematicians of this class ready to deny at first sight that rigorous inferences from the particular to the general were even possible. That they are, in fact, possible is, I suppose, recognized by all who are familiar with the modern work. It will be sufficient here to note that the denial implies, qualitatively, that the process of learning by observation, or experiment, must always lack real cogency. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general • Inferences can be both uncertain and rigorous • Rigor involves adequately specifying uncertainty (??) • For mathematicians trained in deduction, uncertainty arises from a lack of rigor • Induction cannot be rigorous. (I think this!) • #coolkids in the know CONTEXT: Introduction 

This distinction between uncertainty and lack of rigour, which should be familiar to all students of the theory of probability, seems not to be widely understood by those mathematicians who have been trained, as most mathematicians are, almost exclusively in the technique of deductive reasoning; indeed, it would not be surprising or exceptional to find mathematicians of this class ready to deny at first sight that rigorous inferences from the particular to the general were even possible. That they are, in fact, possible is, I suppose, recognized by all who are familiar with the modern work. It will be sufficient here to note that the denial implies, qualitatively, that the process of learning by observation, or experiment, must always lack real cogency. NOTES • Logic not math (??) • Drawing conclusions from looking at figures is inductive reasoning • sample is to particular as population is to general • Inferences can be both uncertain and rigorous. • Rigor involves adequately specifying uncertainty (??) • For mathematicians trained in deduction, uncertainty arises from a lack of rigor • Induction cannot be rigorous. (I think this!) • #coolkids in the know • Learning cannot be perfectly formalized (??) CONTEXT: Introduction 

INDUCTION / DEDUCTION Probability and Likelihood 

Although some uncertain inferences can be rigorously expressed in terms of mathematical probability, it does not follow that mathematical probability is an adequate concept for the rigorous expression of uncertain inferences of every kind. This was at first assumed; but once the distinction between the proposition and its converse is clearly stated, it is seen to be an assumption, and a hazardous one. NOTES CONTEXT: Introduction 

Although some uncertain inferences can be rigorously expressed in terms of mathematical probability, it does not follow that mathematical probability is an adequate concept for the rigorous expression of uncertain inferences of every kind. This was at first assumed; but once the distinction between the proposition and its converse is clearly stated, it is seen to be an assumption, and a hazardous one. NOTES • It is hazardous to assume that probability can model all uncertainties CONTEXT: Introduction 

The inferences of the classical theory of probability are all deductive in character. They are statements about the behaviour of individuals, or samples, or sequences of samples, drawn from populations which are fully known. Even when the theory attempted inferences respecting populations, as in the theory of inverse probability, its method of doing so was to introduce an assumption, or postulate, concerning the population of populations from which the unknown population was supposed to have been drawn at random; and so to bring the problem within the domain of the theory of probability, by making it a deduction from the general to the particular. NOTES • It is hazardous to assume that probability can model all uncertainties CONTEXT: Introduction 

The inferences of the classical theory of probability are all deductive in character. They are statements about the behaviour of individuals, or samples, or sequences of samples, drawn from populations which are fully known. Even when the theory attempted inferences respecting populations, as in the theory of inverse probability, its method of doing so was to introduce an assumption, or postulate, concerning the population of populations from which the unknown population was supposed to have been drawn at random; and so to bring the problem within the domain of the theory of probability, by making it a deduction from the general to the particular. NOTES • It is hazardous to assume that probability can model all uncertainties • Probability is deductive CONTEXT: Introduction 

The inferences of the classical theory of probability are all deductive in character. They are statements about the behaviour of individuals, or samples, or sequences of samples, drawn from populations which are fully known. Even when the theory attempted inferences respecting populations, as in the theory of inverse probability, its method of doing so was to introduce an assumption, or postulate, concerning the population of populations from which the unknown population was supposed to have been drawn at random; and so to bring the problem within the domain of the theory of probability, by making it a deduction from the general to the particular. NOTES • It is hazardous to assume that probability can model all uncertainties • Probability is deductive • Given the prior, Bayesian analysis is deductive CONTEXT: Introduction 

The fact that the concept of probability is adequate for the specification of the nature and extent of uncertainty in these deductive arguments is no guarantee of its adequacy for reasoning of a genuinely inductive kind. If it appears in inductive reasoning, as it has appeared in some cases, we shall welcome it as a familiar friend. More generally, however, a mathematical quantity of a different kind, which I have termed mathematical likelihood, appears to take its place as a measure of rational belief when we are reasoning from the sample to the population. NOTES • It is hazardous to assume that probability can model all uncertainties • Probability is deductive • Given the prior, Bayesian analysis is deductive CONTEXT: Introduction 

The fact that the concept of probability is adequate for the specification of the nature and extent of uncertainty in these deductive arguments is no guarantee of its adequacy for reasoning of a genuinely inductive kind. If it appears in inductive reasoning, as it has appeared in some cases, we shall welcome it as a familiar friend. More generally, however, a mathematical quantity of a different kind, which I have termed mathematical likelihood, appears to take its place as a measure of rational belief when we are reasoning from the sample to the population. NOTES • It is hazardous to assume that probability can model all uncertainties • Probability is deductive • Given the prior, Bayesian analysis is deductive • There’s more to induction than probability CONTEXT: Introduction 

The fact that the concept of probability is adequate for the specification of the nature and extent of uncertainty in these deductive arguments is no guarantee of its adequacy for reasoning of a genuinely inductive kind. If it appears in inductive reasoning, as it has appeared in some cases, we shall welcome it as a familiar friend. More generally, however, a mathematical quantity of a different kind, which I have termed mathematical likelihood, appears to take its place as a measure of rational belief when we are reasoning from the sample to the population. NOTES • It is hazardous to assume that probability can model all uncertainties • Probability is deductive • Given the prior, Bayesian analysis is deductive • There’s more to induction than probability • Mathematical likelihood takes the place of probability for inductive inference CONTEXT: Introduction 

The best use I can make of the short time at my disposal is to show how it is that a consideration of the problem of estimation, without postulating any special significance for the likelihood function, and of course without introducing any such postulate as that needed for inverse probability, does really demonstrate the adequacy of the concept of likelihood for inductive reasoning, in the particular logical situation for which it has been introduced. NOTES • It is hazardous to assume that probability can model all uncertainties • Probability is deductive • Given the prior, Bayesian analysis is deductive • There’s more to induction than probability • Mathematical likelihood takes the place of probability for inductive inference CONTEXT: Introduction 

The best use I can make of the short time at my disposal is to show how it is that a consideration of the problem of estimation, without postulating any special significance for the likelihood function, and of course without introducing any such postulate as that needed for inverse probability, does really demonstrate the adequacy of the concept of likelihood for inductive reasoning, in the particular logical situation for which it has been introduced. NOTES • It is hazardous to assume that probability can model all uncertainties • Probability is deductive • Given the prior, Bayesian analysis is deductive • There’s more to induction than probability • Mathematical likelihood takes the place of probability for inductive inference • Likelihood is adequate for inductive parameter estimation CONTEXT: Introduction 

INDUCTION / DEDUCTION What’s the big deal? 

In considering the future progress of the subject it may be necessary to underline certain distinctions between inductive and deductive reasoning which, if unrecognized, might prove serious obstacles to pure mathematicians trained only in deductive methods, who may be attracted by the novelty and diversity of our subject. NOTES CONTEXT: Conclusions 

In considering the future progress of the subject it may be necessary to underline certain distinctions between inductive and deductive reasoning which, if unrecognized, might prove serious obstacles to pure mathematicians trained only in deductive methods, who may be attracted by the novelty and diversity of our subject. NOTES • “data science” CONTEXT: Conclusions 

In deductive reasoning all knowledge obtainable is already latent in the postulates. Rigour is needed to prevent the successive inferences growing less and less accurate as we proceed. The conclusions are never more accurate than the data. In inductive reasoning we are performing part of the process by which new knowledge is created. The conclusions normally grow more and more accurate as more data are included. It should never be true, though it is still often said, that the conclusions are no more accurate than the data on which they are based. Statistical data are always erroneous, in greater or less degree. The study of inductive reasoning is the study of the embryology of knowledge, of the processes by means of which truth is extracted from its native ore in which it is fused with much error. NOTES • “data science” CONTEXT: Conclusions 

In deductive reasoning all knowledge obtainable is already latent in the postulates. Rigour is needed to prevent the successive inferences growing less and less accurate as we proceed. The conclusions are never more accurate than the data. In inductive reasoning we are performing part of the process by which new knowledge is created. The conclusions normally grow more and more accurate as more data are included. It should never be true, though it is still often said, that the conclusions are no more accurate than the data on which they are based. Statistical data are always erroneous, in greater or less degree. The study of inductive reasoning is the study of the embryology of knowledge, of the processes by means of which truth is extracted from its native ore in which it is fused with much error. NOTES • “data science” • DE-duction is RE-duction CONTEXT: Conclusions 

In deductive reasoning all knowledge obtainable is already latent in the postulates. Rigour is needed to prevent the successive inferences growing less and less accurate as we proceed. The conclusions are never more accurate than the data. In inductive reasoning we are performing part of the process by which new knowledge is created. The conclusions normally grow more and more accurate as more data are included. It should never be true, though it is still often said, that the conclusions are no more accurate than the data on which they are based. Statistical data are always erroneous, in greater or less degree. The study of inductive reasoning is the study of the embryology of knowledge, of the processes by means of which truth is extracted from its native ore in which it is fused with much error. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? CONTEXT: Conclusions 

In deductive reasoning all knowledge obtainable is already latent in the postulates. Rigour is needed to prevent the successive inferences growing less and less accurate as we proceed. The conclusions are never more accurate than the data. In inductive reasoning we are performing part of the process by which new knowledge is created. The conclusions normally grow more and more accurate as more data are included. It should never be true, though it is still often said, that the conclusions are no more accurate than the data on which they are based. Statistical data are always erroneous, in greater or less degree. The study of inductive reasoning is the study of the embryology of knowledge, of the processes by means of which truth is extracted from its native ore in which it is fused with much error. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? • Inductive conclusions are more than deductive consequences of data + program CONTEXT: Conclusions 

Secondly, rigour, as understood in deductive mathematics, is not enough. In deductive reasoning, conclusions based on any chosen few of the postulates accepted need only mathematical rigour to guarantee their truth. All statisticians know that data are falsified if only a selected part is used. Inductive reasoning cannot aim at a truth that is less than the whole truth. Our conclusions must be warranted by the whole of the data, since less than the whole may be to any degree misleading. This, of course, is no reason against the use of absolutely precise forms of statement when these are available. It is only a warning to those who may be tempted to think that the particular precise code of mathematical statements in which they have been drilled at College is a substitute for the use of reasoning powers, which mankind has probably possessed since prehistoric times, and in which, as the history of the theory of probability shows, the process of codification is still incomplete. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? • Inductive conclusions are more than deductive consequences of data + program CONTEXT: Conclusions 

Secondly, rigour, as understood in deductive mathematics, is not enough. In deductive reasoning, conclusions based on any chosen few of the postulates accepted need only mathematical rigour to guarantee their truth. All statisticians know that data are falsified if only a selected part is used. Inductive reasoning cannot aim at a truth that is less than the whole truth. Our conclusions must be warranted by the whole of the data, since less than the whole may be to any degree misleading. This, of course, is no reason against the use of absolutely precise forms of statement when these are available. It is only a warning to those who may be tempted to think that the particular precise code of mathematical statements in which they have been drilled at College is a substitute for the use of reasoning powers, which mankind has probably possessed since prehistoric times, and in which, as the history of the theory of probability shows, the process of codification is still incomplete. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? • Inductive conclusions are more than deductive consequences of data + program • Mathematical rigor is not sufficient for induction CONTEXT: Conclusions 

Secondly, rigour, as understood in deductive mathematics, is not enough. In deductive reasoning, conclusions based on any chosen few of the postulates accepted need only mathematical rigour to guarantee their truth. All statisticians know that data are falsified if only a selected part is used. Inductive reasoning cannot aim at a truth that is less than the whole truth. Our conclusions must be warranted by the whole of the data, since less than the whole may be to any degree misleading. This, of course, is no reason against the use of absolutely precise forms of statement when these are available. It is only a warning to those who may be tempted to think that the particular precise code of mathematical statements in which they have been drilled at College is a substitute for the use of reasoning powers, which mankind has probably possessed since prehistoric times, and in which, as the history of the theory of probability shows, the process of codification is still incomplete. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? • Inductive conclusions are more than deductive consequences of data + program • Mathematical rigor is not sufficient for induction • Meaning: not random? CONTEXT: Conclusions 

Secondly, rigour, as understood in deductive mathematics, is not enough. In deductive reasoning, conclusions based on any chosen few of the postulates accepted need only mathematical rigour to guarantee their truth. All statisticians know that data are falsified if only a selected part is used. Inductive reasoning cannot aim at a truth that is less than the whole truth. Our conclusions must be warranted by the whole of the data, since less than the whole may be to any degree misleading. This, of course, is no reason against the use of absolutely precise forms of statement when these are available. It is only a warning to those who may be tempted to think that the particular precise code of mathematical statements in which they have been drilled at College is a substitute for the use of reasoning powers, which mankind has probably possessed since prehistoric times, and in which, as the history of the theory of probability shows, the process of codification is still incomplete. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? • Inductive conclusions are more than deductive consequences of data + program • Mathematical rigor is not sufficient for induction • Meaning: not random? • Induction diversity principle CONTEXT: Conclusions 

Secondly, rigour, as understood in deductive mathematics, is not enough. In deductive reasoning, conclusions based on any chosen few of the postulates accepted need only mathematical rigour to guarantee their truth. All statisticians know that data are falsified if only a selected part is used. Inductive reasoning cannot aim at a truth that is less than the whole truth. Our conclusions must be warranted by the whole of the data, since less than the whole may be to any degree misleading. This, of course, is no reason against the use of absolutely precise forms of statement when these are available. It is only a warning to those who may be tempted to think that the particular precise code of mathematical statements in which they have been drilled at College is a substitute for the use of reasoning powers, which mankind has probably possessed since prehistoric times, and in which, as the history of the theory of probability shows, the process of codification is still incomplete. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? • Inductive conclusions are more than deductive consequences of data + program • Mathematical rigor is not sufficient for induction • Meaning: not random? • Induction diversity principle • Humans have innate inductive reasoning skills that extend beyond deduction CONTEXT: Conclusions 

Secondly, rigour, as understood in deductive mathematics, is not enough. In deductive reasoning, conclusions based on any chosen few of the postulates accepted need only mathematical rigour to guarantee their truth. All statisticians know that data are falsified if only a selected part is used. Inductive reasoning cannot aim at a truth that is less than the whole truth. Our conclusions must be warranted by the whole of the data, since less than the whole may be to any degree misleading. This, of course, is no reason against the use of absolutely precise forms of statement when these are available. It is only a warning to those who may be tempted to think that the particular precise code of mathematical statements in which they have been drilled at College is a substitute for the use of reasoning powers, which mankind has probably possessed since prehistoric times, and in which, as the history of the theory of probability shows, the process of codification is still incomplete. NOTES • “data science” • DE-duction is RE-duction • Less rigor is less accurate? • Inductive conclusions are more than deductive consequences of data + program • Mathematical rigor is not sufficient for induction • Meaning: not random? • Induction diversity principle • Humans have innate inductive reasoning skills that extend beyond deduction • More to do! CONTEXT: Conclusions 

INDUCTION / DEDUCTION A Critical Discussion 

Dr. Isserlis Man is an inductive animal; we all generalize from the particular to the general; in all branches of science, and not only in statistics, it is the business of those of us who have devoted some attention to our own branch of the subject, to try and act as guides to our followers in preventing rash generalization. Speaking as a mathematician as well as a statistician, I find it rather difficult to follow the paragraphs on p. 39 of the paper where Professor Fisher tells us that mathematicians trained in deductive methods are apt to forget that rigorous inferences from the particular to the general are even possible. I do not think that is the case with the ordinary mathematician. It may be that in mathematical analysis the fundamental inductions on which the analysis rests are rather remote, but they are there all right, and no mathematician may proceed safely with his work unless he is strongly aware of their existence. NOTES CONTEXT: Discussion 

Dr. Isserlis Man is an inductive animal; we all generalize from the particular to the general; in all branches of science, and not only in statistics, it is the business of those of us who have devoted some attention to our own branch of the subject, to try and act as guides to our followers in preventing rash generalization. Speaking as a mathematician as well as a statistician, I find it rather difficult to follow the paragraphs on p. 39 of the paper where Professor Fisher tells us that mathematicians trained in deductive methods are apt to forget that rigorous inferences from the particular to the general are even possible. I do not think that is the case with the ordinary mathematician. It may be that in mathematical analysis the fundamental inductions on which the analysis rests are rather remote, but they are there all right, and no mathematician may proceed safely with his work unless he is strongly aware of their existence. NOTES • Our nature is to induce CONTEXT: Discussion 

Dr. Isserlis Man is an inductive animal; we all generalize from the particular to the general; in all branches of science, and not only in statistics, it is the business of those of us who have devoted some attention to our own branch of the subject, to try and act as guides to our followers in preventing rash generalization. Speaking as a mathematician as well as a statistician, I find it rather difficult to follow the paragraphs on p. 39 of the paper where Professor Fisher tells us that mathematicians trained in deductive methods are apt to forget that rigorous inferences from the particular to the general are even possible. I do not think that is the case with the ordinary mathematician. It may be that in mathematical analysis the fundamental inductions on which the analysis rests are rather remote, but they are there all right, and no mathematician may proceed safely with his work unless he is strongly aware of their existence. NOTES • Our nature is to induce • Math rests on axioms CONTEXT: Discussion 

Professor Wolf (the logician) PROFESSOR WOLF thanked the President for inviting him to listen to this paper and the very instructive discussion, and for allowing him to take part in the discussion. He was not a mathematician, nor a statistician, and he could not, therefore, be expected to make any contribution towards the mathematics of the paper, but he had all his life been interested in the study of scientific method. Unfortunately there were very few men of science who had ever seriously thought about the basic methods and principles of science, or, at all events, who had published their reflections upon the principles which underlay their scientific investigations. Therefore when he came across men of science who had the courage to do that kind of thing, he wanted to thank them very gratefully, and he did thank Professor Fisher. NOTES • Our nature is to induce • Math rests on axioms Abraham Wolf (1876-1948) CONTEXT: Discussion 

Professor Wolf (the logician) PROFESSOR WOLF thanked the President for inviting him to listen to this paper and the very instructive discussion, and for allowing him to take part in the discussion. He was not a mathematician, nor a statistician, and he could not, therefore, be expected to make any contribution towards the mathematics of the paper, but he had all his life been interested in the study of scientific method. Unfortunately there were very few men of science who had ever seriously thought about the basic methods and principles of science, or, at all events, who had published their reflections upon the principles which underlay their scientific investigations. Therefore when he came across men of science who had the courage to do that kind of thing, he wanted to thank them very gratefully, and he did thank Professor Fisher. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous Abraham Wolf (1876-1948) CONTEXT: Discussion 

Professor Wolf (the logician) … he would like to ask what was the net result of these estimates to be? Were these estimates finally to be merely of a subjective value, or were they intended to have an objective, scientific character? What he meant by this would be obvious if he took the case of the theory of probability. So far as he was concerned, he had maintained for many years that there were both types of estimates of probability, the deductive and the inductive calculation of probability; but from a scientific point of view he believed that the real value lay in the knowledge of the frequencies. In inductive calculations one started from the sample frequencies, and deduced their probabilities. In the deductive calculations one started from the a priori probabilities, and from these it was possible, more or less securely, to deduce the probable frequencies. But, in either case, the real scientific value lay in the frequencies rather than in the probabilities. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous CONTEXT: Discussion 

Professor Wolf (the logician) … he would like to ask what was the net result of these estimates to be? Were these estimates finally to be merely of a subjective value, or were they intended to have an objective, scientific character? What he meant by this would be obvious if he took the case of the theory of probability. So far as he was concerned, he had maintained for many years that there were both types of estimates of probability, the deductive and the inductive calculation of probability; but from a scientific point of view he believed that the real value lay in the knowledge of the frequencies. In inductive calculations one started from the sample frequencies, and deduced their probabilities. In the deductive calculations one started from the a priori probabilities, and from these it was possible, more or less securely, to deduce the probable frequencies. But, in either case, the real scientific value lay in the frequencies rather than in the probabilities. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? CONTEXT: Discussion 

Professor Wolf (the logician) … he would like to ask what was the net result of these estimates to be? Were these estimates finally to be merely of a subjective value, or were they intended to have an objective, scientific character? What he meant by this would be obvious if he took the case of the theory of probability. So far as he was concerned, he had maintained for many years that there were both types of estimates of probability, the deductive and the inductive calculation of probability; but from a scientific point of view he believed that the real value lay in the knowledge of the frequencies. In inductive calculations one started from the sample frequencies, and deduced their probabilities. In the deductive calculations one started from the a priori probabilities, and from these it was possible, more or less securely, to deduce the probable frequencies. But, in either case, the real scientific value lay in the frequencies rather than in the probabilities. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective CONTEXT: Discussion 

Professor Wolf (the logician) Estimates of probability seemed to be more of psychological, rather than of general scientific, importance. When he compared different fractions of probability as the measure of what his rational belief ought to be, he found it impossible to adjust his belief to these different fractions. Even subjectively, therefore, calculations of probability seemed unimportant. He could not find any real, scientific, or strictly objective significance in probabilities as such. When he said that measures of probability were a matter of psychological or subjective interest, he realized, of course, that they were logical in character, and therefore, in a secondary sense, objective, that is to say, they were not capriciously subjective; but nevertheless it remained true that he did not find it within his competence to adjust his degree of rational belief to the different requirements of the different estimates of probability. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective CONTEXT: Discussion 

Professor Wolf (the logician) Estimates of probability seemed to be more of psychological, rather than of general scientific, importance. When he compared different fractions of probability as the measure of what his rational belief ought to be, he found it impossible to adjust his belief to these different fractions. Even subjectively, therefore, calculations of probability seemed unimportant. He could not find any real, scientific, or strictly objective significance in probabilities as such. When he said that measures of probability were a matter of psychological or subjective interest, he realized, of course, that they were logical in character, and therefore, in a secondary sense, objective, that is to say, they were not capriciously subjective; but nevertheless it remained true that he did not find it within his competence to adjust his degree of rational belief to the different requirements of the different estimates of probability. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective • Hard to quantify belief! CONTEXT: Discussion 

Professor Wolf (the logician) Estimates of probability seemed to be more of psychological, rather than of general scientific, importance. When he compared different fractions of probability as the measure of what his rational belief ought to be, he found it impossible to adjust his belief to these different fractions. Even subjectively, therefore, calculations of probability seemed unimportant. He could not find any real, scientific, or strictly objective significance in probabilities as such. When he said that measures of probability were a matter of psychological or subjective interest, he realized, of course, that they were logical in character, and therefore, in a secondary sense, objective, that is to say, they were not capriciously subjective; but nevertheless it remained true that he did not find it within his competence to adjust his degree of rational belief to the different requirements of the different estimates of probability. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective • Hard to quantify belief! • Probability is not capriciously subjective CONTEXT: Discussion 

Professor Wolf (the logician) Professor Wolf said he did not propose to add any comments on the more limited problems with which the lecturer had dealt. He was more interested in the wider problem suggested by the title of Dr. Fisher's paper, namely, the general problem of the logic of induction. It was gratifying to him personally to find that Professor Fisher repudiated the old idea that the whole of induction was based on the calculation of probability. Two or three decades ago that was more or less the prevalent conception of induction. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective • Hard to quantify belief! • Probability is not capriciously subjective CONTEXT: Discussion 

Professor Wolf (the logician) Professor Wolf said he did not propose to add any comments on the more limited problems with which the lecturer had dealt. He was more interested in the wider problem suggested by the title of Dr. Fisher's paper, namely, the general problem of the logic of induction. It was gratifying to him personally to find that Professor Fisher repudiated the old idea that the whole of induction was based on the calculation of probability. Two or three decades ago that was more or less the prevalent conception of induction. NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective • Hard to quantify belief! • Probability is not capriciously subjective • Induction is not based on probability calculations CONTEXT: Discussion 

Professor Wolf (the logician) With regard to some of the misapprehensions which underlay the older conception of the statistical basis of induction, it was not quite clear whether Professor Fisher was entirely free from them, in spite of the fact that in one place he distinctly repudiated them. The storm-centre lay very largely in the conception of mathematics and of its place in science. There was the familiar idea that pure mathematics was entirely deductive; and a great many people held that view. The conception that probability was at the base of all induction was largely the progeny of this conception of pure mathematics. The idea underlying that belief was that pure mathematics was exact and absolutely reliable; it did not make any assumption of an inductive character, and was therefore qualified to serve as a basis of inductive inference. Professor Wolf was very doubtful about this. He did not believe that pure mathematics was purely deductive. There was induction in mathematics, but it was slurred over. Owing perhaps to bad teaching, encouragement had been given to the assumption that mathematics was all deductive, and not at all inductive. How was it that mathematics has thus come to be associated solely with deduction? NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective • Hard to quantify belief! • Probability is not capriciously subjective • Induction is not based on probability calculations CONTEXT: Discussion 

Professor Wolf (the logician) With regard to some of the misapprehensions which underlay the older conception of the statistical basis of induction, it was not quite clear whether Professor Fisher was entirely free from them, in spite of the fact that in one place he distinctly repudiated them. The storm-centre lay very largely in the conception of mathematics and of its place in science. There was the familiar idea that pure mathematics was entirely deductive; and a great many people held that view. The conception that probability was at the base of all induction was largely the progeny of this conception of pure mathematics. The idea underlying that belief was that pure mathematics was exact and absolutely reliable; it did not make any assumption of an inductive character, and was therefore qualified to serve as a basis of inductive inference. Professor Wolf was very doubtful about this. He did not believe that pure mathematics was purely deductive. There was induction in mathematics, but it was slurred over. Owing perhaps to bad teaching, encouragement had been given to the assumption that mathematics was all deductive, and not at all inductive. How was it that mathematics has thus come to be associated solely with deduction? NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective • Hard to quantify belief! • Probability is not capriciously subjective • Induction is not based on probability calculations • Axioms require induction CONTEXT: Discussion 

Professor Wolf (the logician) With regard to some of the misapprehensions which underlay the older conception of the statistical basis of induction, it was not quite clear whether Professor Fisher was entirely free from them, in spite of the fact that in one place he distinctly repudiated them. The storm-centre lay very largely in the conception of mathematics and of its place in science. There was the familiar idea that pure mathematics was entirely deductive; and a great many people held that view. The conception that probability was at the base of all induction was largely the progeny of this conception of pure mathematics. The idea underlying that belief was that pure mathematics was exact and absolutely reliable; it did not make any assumption of an inductive character, and was therefore qualified to serve as a basis of inductive inference. Professor Wolf was very doubtful about this. He did not believe that pure mathematics was purely deductive. There was induction in mathematics, but it was slurred over. Owing perhaps to bad teaching, encouragement had been given to the assumption that mathematics was all deductive, and not at all inductive. How was it that mathematics has thus come to be associated solely with deduction? NOTES • Our nature is to induce • Math rests on axioms • Reflecting on foundations is courageous • Did Fisher intend to be objective or subjective? • Frequencies are scientific and probability is subjective • Hard to quantify belief! • Probability is not capriciously subjective • Induction is not based on probability calculations • Axioms require induction • Why is math associated with deduction? CONTEXT: Discussion 

Professor Wolf (the logician) The misapprehension was probably due to three contributory factors. (1) The idea was upheld partly by Descartes, who played such an important rôle in the whole development of modern mathematics that his word was accepted without challenge. But if one studied Descartes’ use of the term “deduction” it would be seen that he did not use it in the ordinary sense of inference from general propositions, definitely accepted, or assumed provisionally; he used it in a much more complicated sense, which included a good deal of induction. NOTES • Why is math associated with deduction? 1) Decartes! CONTEXT: Discussion 

Professor Wolf (the logician) (2) People were still frequently using the term “deduction” not in its ordinary sense--“inference from the general to the particular or to the less general”--but for inference of any and every kind. A common phrase was, “What deductions do you draw from these facts?” Deductions (properly so called) were not drawn from facts; “inferences” was the word that should be used in such contexts. There was thus a very common use of the term “deduction” for “inference”; and people did not always realize that they were talking about inference in general, and not about deduction in particular. NOTES • Why is math associated with deduction? 1) Decartes! 2) People used words imprecisely CONTEXT: Discussion 

Professor Wolf (the logician) (3) A third point was perhaps even more important. Mathematicians and scientists generally did not realize sufficiently that in what was called “inductive inference” there was nearly always a moment, or stage, which was deductive, namely, the stage where the hypothesis had to be verified, and this was done by application to suitable cases of the hypothesis, which was a general statement accepted as possibly true. That stage was purely deductive, yet the investigation as a whole was essentially inductive. It was not sufficiently realized that although there might be deductions without inductions, there could not be--- except in very rare cases---induction without a deductive moment or stage. In mathematics, no doubt, the deductive moment loomed very large, and so people jumped to the conclusion that the whole of mathematics was deductive. Professor Wolf did not accept that view; and as soon as it was realized that even mathematics was partly inductive, one could see for oneself that mathematics, or any part of it, could not be made the logical basis of all other forms of induction. NOTES • Why is math associated with deduction? 1) Decartes! 2) People used words imprecisely 3) People jumped to conclusions CONTEXT: Discussion 

Professor Wolf (the logician) (3) A third point was perhaps even more important. Mathematicians and scientists generally did not realize sufficiently that in what was called “inductive inference” there was nearly always a moment, or stage, which was deductive, namely, the stage where the hypothesis had to be verified, and this was done by application to suitable cases of the hypothesis, which was a general statement accepted as possibly true. That stage was purely deductive, yet the investigation as a whole was essentially inductive. It was not sufficiently realized that although there might be deductions without inductions, there could not be--- except in very rare cases---induction without a deductive moment or stage. In mathematics, no doubt, the deductive moment loomed very large, and so people jumped to the conclusion that the whole of mathematics was deductive. Professor Wolf did not accept that view; and as soon as it was realized that even mathematics was partly inductive, one could see for oneself that mathematics, or any part of it, could not be made the logical basis of all other forms of induction. NOTES • Why is math associated with deduction? 1) Decartes! 2) People used words imprecisely 3) People jumped to conclusions • Axioms! CONTEXT: Discussion 

Professor Wolf (the logician) To pass to another point, Professor Wolf sometimes wondered whether the tendency to exaggerate the importance of mathematics, and especially the theory of probability, in inductive science was not due to a very large extent to the disbelief, on the part of the exponents, in the possibility of induction altogether; whether, in fact, it was not due to their conception that not only was so-called “probability” a subjective matter, but that the whole of scientific inference was mainly the subjective play of the human mind attempting to amuse itself, or to satisfy itself, by means of man-made conjectures which might not reflect reality at all. Mr. Bertrand Russell in one of his latest books has made this idea perfectly clear. He has said that, for all that was known, natural phenomena might contain no order at all, and that it was only the cleverness of mathematicians which imposed on Nature an appearance of order. NOTES • Why is math associated with deduction? 1) Decartes! 2) People used words imprecisely 3) People jumped to conclusions • Axioms! CONTEXT: Discussion 

Professor Wolf (the logician) To pass to another point, Professor Wolf sometimes wondered whether the tendency to exaggerate the importance of mathematics, and especially the theory of probability, in inductive science was not due to a very large extent to the disbelief, on the part of the exponents, in the possibility of induction altogether; whether, in fact, it was not due to their conception that not only was so-called “probability” a subjective matter, but that the whole of scientific inference was mainly the subjective play of the human mind attempting to amuse itself, or to satisfy itself, by means of man-made conjectures which might not reflect reality at all. Mr. Bertrand Russell in one of his latest books has made this idea perfectly clear. He has said that, for all that was known, natural phenomena might contain no order at all, and that it was only the cleverness of mathematicians which imposed on Nature an appearance of order. NOTES • Why is math associated with deduction? 1) Decartes! 2) People used words imprecisely 3) People jumped to conclusions • Axioms! • Some say: math is so important because induction doesn’t exist CONTEXT: Discussion 

Professor Wolf (the logician) Although he was not a mathematician, Professor Wolf did not believe that Mr. Russell could discover a formula showing order among phenomena utterly disordered. Here was a tendency to exaggerate the importance of mathematics, coupled with scepticism as to the real objective value of science---a scepticism as to the real existence of orderliness among natural phenomena. To some extent the same tendency might be found in Professor Karl Pearson. On looking at his Grammar of Science it would be seen how he was smitten with Kantian philosophy interpreted in such a way as to make all knowledge the invention or creation of the mind, so that the orderliness that was found in Nature was simply the orderliness which the human mind imposed upon natural phenomena. NOTES • Why is math associated with deduction? 1) Decartes! 2) People used words imprecisely 3) People jumped to conclusions • Axioms! • Some say: math is so important because induction doesn’t exist CONTEXT: Discussion 

Professor Wolf (the logician) Although he was not a mathematician, Professor Wolf did not believe that Mr. Russell could discover a formula showing order among phenomena utterly disordered. Here was a tendency to exaggerate the importance of mathematics, coupled with scepticism as to the real objective value of science---a scepticism as to the real existence of orderliness among natural phenomena. To some extent the same tendency might be found in Professor Karl Pearson. On looking at his Grammar of Science it would be seen how he was smitten with Kantian philosophy interpreted in such a way as to make all knowledge the invention or creation of the mind, so that the orderliness that was found in Nature was simply the orderliness which the human mind imposed upon natural phenomena. NOTES • Why is math associated with deduction? 1) Decartes! 2) People used words imprecisely 3) People jumped to conclusions • Axioms! • Some say: math is so important because induction doesn’t exist • Some say: maybe it’s all in our heads CONTEXT: Discussion 

Fisher’s reply In reply to Professor Wolf I should probably have explained that, following Bayes, and, I believe, most of the early writers, but unlike Laplace, and others influenced by him in the nineteenth century, I mean by mathematical probability only that objective quality of the individual which corresponds to frequency in the population, of which the individual is spoken of as a typical member. It is of great interest that Professor Wolf had concluded long ago that the concept of probability was inadequate as a basis for inductive reasoning. I believe we may add that, in so far as an induction can be cogent, it must be capable of rigorous mathematical justification, and that the concept of mathematical likelihood makes this possible in the important logical situation presented by problems of estimation. NOTES CONTEXT: Fisher’s reply 

Fisher’s reply In reply to Professor Wolf I should probably have explained that, following Bayes, and, I believe, most of the early writers, but unlike Laplace, and others influenced by him in the nineteenth century, I mean by mathematical probability only that objective quality of the individual which corresponds to frequency in the population, of which the individual is spoken of as a typical member. It is of great interest that Professor Wolf had concluded long ago that the concept of probability was inadequate as a basis for inductive reasoning. I believe we may add that, in so far as an induction can be cogent, it must be capable of rigorous mathematical justification, and that the concept of mathematical likelihood makes this possible in the important logical situation presented by problems of estimation. NOTES • Probability is objective as frequency CONTEXT: Fisher’s reply 

Fisher’s reply In reply to Professor Wolf I should probably have explained that, following Bayes, and, I believe, most of the early writers, but unlike Laplace, and others influenced by him in the nineteenth century, I mean by mathematical probability only that objective quality of the individual which corresponds to frequency in the population, of which the individual is spoken of as a typical member. It is of great interest that Professor Wolf had concluded long ago that the concept of probability was inadequate as a basis for inductive reasoning. I believe we may add that, in so far as an induction can be cogent, it must be capable of rigorous mathematical justification, and that the concept of mathematical likelihood makes this possible in the important logical situation presented by problems of estimation. NOTES • Probability is objective as frequency • Likelihood makes induction cogent in parameter estimation CONTEXT: Fisher’s reply 

Fisher’s reply I did not suggest that mathematics could be entirely deductive, but that the current training of pure mathematicians gave them no experience of the rigorous handling of inductive processes. Professor Wolf expresses my thought well when he says “there is induction in mathematics, but it is slurred over,” but I should myself prefer to say “in mathematical applications,” for some mathematical reasoning is purely deductive. With Professor Wolf's third point I am inclined to disagree. He says: “As soon as it is realized that even mathematics was partly inductive, one could see for oneself that mathematics, or any part of it, could not be made the logical basis for all other forms of induction.” This suggests that mathematics can be made the logical basis of deductive reasoning, but I doubt if this is what Professor Wolf means. I should rather say that all reasoning may properly be called mathematical, in so far as it is concise, cogent, and of general application. In this view mathematics is always no more than a means of efficient reasoning, and never attempts to provide its logical basis. NOTES • Probability is objective as frequency • Likelihood makes induction cogent in parameter estimation CONTEXT: Fisher’s reply 

Fisher’s reply I did not suggest that mathematics could be entirely deductive, but that the current training of pure mathematicians gave them no experience of the rigorous handling of inductive processes. Professor Wolf expresses my thought well when he says “there is induction in mathematics, but it is slurred over,” but I should myself prefer to say “in mathematical applications,” for some mathematical reasoning is purely deductive. With Professor Wolf's third point I am inclined to disagree. He says: “As soon as it is realized that even mathematics was partly inductive, one could see for oneself that mathematics, or any part of it, could not be made the logical basis for all other forms of induction.” This suggests that mathematics can be made the logical basis of deductive reasoning, but I doubt if this is what Professor Wolf means. I should rather say that all reasoning may properly be called mathematical, in so far as it is concise, cogent, and of general application. In this view mathematics is always no more than a means of efficient reasoning, and never attempts to provide its logical basis.. NOTES • Probability is objective as frequency • Likelihood makes induction cogent in parameter estimation • I think Fisher is misreading Wolf CONTEXT: Fisher’s reply 

Fisher’s reply I did not suggest that mathematics could be entirely deductive, but that the current training of pure mathematicians gave them no experience of the rigorous handling of inductive processes. Professor Wolf expresses my thought well when he says “there is induction in mathematics, but it is slurred over,” but I should myself prefer to say “in mathematical applications,” for some mathematical reasoning is purely deductive. With Professor Wolf's third point I am inclined to disagree. He says: “As soon as it is realized that even mathematics was partly inductive, one could see for oneself that mathematics, or any part of it, could not be made the logical basis for all other forms of induction.” This suggests that mathematics can be made the logical basis of deductive reasoning, but I doubt if this is what Professor Wolf means. I should rather say that all reasoning may properly be called mathematical, in so far as it is concise, cogent, and of general application. In this view mathematics is always no more than a means of efficient reasoning, and never attempts to provide its logical basis. NOTES • Probability is objective as frequency • Likelihood makes induction cogent in parameter estimation • I think Fisher is misreading Wolf • I don’t think any of these three things make reasoning mathematical CONTEXT: Fisher’s reply 

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A PHILOSOPHY OF FINITE SAMPLES [BACK-UP SLIDES] 

We are now in a position to consider the real problem of finite samples. For any method of estimation has its own characteristic distribution of errors, not now necessarily normal, and therefore its own intrinsic accuracy. Consequently, the amount of information which it extracts from the data is calculable, and it is possible to compare the merits of different estimates, even though they all satisfy the criterion of efficiency in the limit for large samples. It is obvious, too, that in introducing the concept of quantity of information we do not want merely to be giving an arbitrary name to a calculable quantity, but must be prepared to justify the term employed, in relation to what common sense requires, if the term is to be appropriate, and serviceable as a tool for thinking. The mathematical consequences of identifying, as I propose, the intrinsic accuracy of the error curve, with the amount of information extracted, may therefore be summarized specifically in order that we may judge by our pre-mathematical common sense whether they are the properties it ought to have. NOTES • Compare estimates using information criteria [BACK-UP SLIDES] 

We are now in a position to consider the real problem of finite samples. For any method of estimation has its own characteristic distribution of errors, not now necessarily normal, and therefore its own intrinsic accuracy. Consequently, the amount of information which it extracts from the data is calculable, and it is possible to compare the merits of different estimates, even though they all satisfy the criterion of efficiency in the limit for large samples. It is obvious, too, that in introducing the concept of quantity of information we do not want merely to be giving an arbitrary name to a calculable quantity, but must be prepared to justify the term employed, in relation to what common sense requires, if the term is to be appropriate, and serviceable as a tool for thinking. The mathematical consequences of identifying, as I propose, the intrinsic accuracy of the error curve, with the amount of information extracted, may therefore be summarized specifically in order that we may judge by our pre-mathematical common sense whether they are the properties it ought to have. NOTES • Compare estimates using information criteria • Common sense will assess the value of information! [BACK-UP SLIDES] 

First, then, when the probabilities of the different kinds of observation which can be made are all independent of a particular parameter, the observations will supply no information about the parameter. … In certain cases estimates are shown to exist such that, when they are given, the distributions of all other estimates are independent of the parameter required. Such estimates, which are called sufficient, contain, even from finite samples, the whole of the information supplied by the data. Thirdly, the information extracted by an estimate can never exceed the total quantity present in the data. And, fourthly, statistically independent observations supply amounts of information which are additive. One could, therefore, develop a mathematical theory of quantity of information from these properties as postulates, and this would be the normal mathematical procedure. It is, perhaps, only a personal preference that I am more inclined to examine the quantity as it emerges from mathematical investigations and to judge of its utility by the free use of common sense, rather than to impose it by a formal definition. NOTES • Compare estimates using information criteria • Common sense will assess the value of information! - uninformative data [BACK-UP SLIDES] 

First, then, when the probabilities of the different kinds of observation which can be made are all independent of a particular parameter, the observations will supply no information about the parameter. … In certain cases estimates are shown to exist such that, when they are given, the distributions of all other estimates are independent of the parameter required. Such estimates, which are called sufficient, contain, even from finite samples, the whole of the information supplied by the data. Thirdly, the information extracted by an estimate can never exceed the total quantity present in the data. And, fourthly, statistically independent observations supply amounts of information which are additive. One could, therefore, develop a mathematical theory of quantity of information from these properties as postulates, and this would be the normal mathematical procedure. It is, perhaps, only a personal preference that I am more inclined to examine the quantity as it emerges from mathematical investigations and to judge of its utility by the free use of common sense, rather than to impose it by a formal definition. NOTES • Compare estimates using information criteria • Common sense will assess the value of information! - uninformative data - sufficiency [BACK-UP SLIDES] 

First, then, when the probabilities of the different kinds of observation which can be made are all independent of a particular parameter, the observations will supply no information about the parameter. … In certain cases estimates are shown to exist such that, when they are given, the distributions of all other estimates are independent of the parameter required. Such estimates, which are called sufficient, contain, even from finite samples, the whole of the information supplied by the data. Thirdly, the information extracted by an estimate can never exceed the total quantity present in the data. And, fourthly, statistically independent observations supply amounts of information which are additive. One could, therefore, develop a mathematical theory of quantity of information from these properties as postulates, and this would be the normal mathematical procedure. It is, perhaps, only a personal preference that I am more inclined to examine the quantity as it emerges from mathematical investigations and to judge of its utility by the free use of common sense, rather than to impose it by a formal definition. NOTES • Compare estimates using information criteria • Common sense will assess the value of information! - uninformative data - sufficiency - max information [BACK-UP SLIDES] 

First, then, when the probabilities of the different kinds of observation which can be made are all independent of a particular parameter, the observations will supply no information about the parameter. … In certain cases estimates are shown to exist such that, when they are given, the distributions of all other estimates are independent of the parameter required. Such estimates, which are called sufficient, contain, even from finite samples, the whole of the information supplied by the data. Thirdly, the information extracted by an estimate can never exceed the total quantity present in the data. And, fourthly, statistically independent observations supply amounts of information which are additive. One could, therefore, develop a mathematical theory of quantity of information from these properties as postulates, and this would be the normal mathematical procedure. It is, perhaps, only a personal preference that I am more inclined to examine the quantity as it emerges from mathematical investigations and to judge of its utility by the free use of common sense, rather than to impose it by a formal definition. NOTES • Compare estimates using information criteria • Common sense will assess the value of information! - uninformative data - sufficiency - max information - additive information [BACK-UP SLIDES] 

First, then, when the probabilities of the different kinds of observation which can be made are all independent of a particular parameter, the observations will supply no information about the parameter. … In certain cases estimates are shown to exist such that, when they are given, the distributions of all other estimates are independent of the parameter required. Such estimates, which are called sufficient, contain, even from finite samples, the whole of the information supplied by the data. Thirdly, the information extracted by an estimate can never exceed the total quantity present in the data. And, fourthly, statistically independent observations supply amounts of information which are additive. One could, therefore, develop a mathematical theory of quantity of information from these properties as postulates, and this would be the normal mathematical procedure. It is, perhaps, only a personal preference that I am more inclined to examine the quantity as it emerges from mathematical investigations and to judge of its utility by the free use of common sense, rather than to impose it by a formal definition. NOTES • Compare estimates using information criteria • Common sense will assess the value of information! - uninformative data - sufficiency - max information - additive information • Freedom from formalism! [BACK-UP SLIDES]